A024843 a(n) = least m such that if r and s in {1/1, 1/2, 1/3, ..., 1/n} satisfy r < s, then r < k/m < (k+3)/m < s for some integer k.

9, 23, 43, 69, 101, 139, 183, 233, 289, 361, 431, 518, 601, 703, 799, 916, 1025, 1157, 1279, 1426, 1561, 1723, 1871, 2048, 2209, 2401, 2601, 2783, 2998, 3221, 3423, 3661, 3907, 4129, 4390, 4659, 4901, 5185, 5477, 5739, 6046, 6361, 6643, 6973, 7311, 7613, 7966, 8327, 8649

Offset: 2

Author

Clark Kimberling

Keywords

• nonn

Comments

For a guide to related sequences, see A001000. - Clark Kimberling, Aug 08 2012

Examples

			Using the terminology introduced at A001000, the 4th separator of the set {1/3, 1/2, 1} is a(3) = 23, since 1/3 < 8/23 < 11/23 < 1/2 < 12/23 < 15/23 < 1 and 23 is the least m for which 1/3, 1/2, 1 are thus separated using numbers k/m. - _Clark Kimberling_, Aug 08 2012
		

Links

• Clark Kimberling, Table of n, a(n) for n = 2..100

Crossrefs

Cf. A001000.

Programs

Mathematica

leastSeparatorS[seq_, s_] := Module[{n = 1},
Table[While[Or @@ (Ceiling[n #1[[1]]] <
s + 1 + Floor[n #1[[2]]] &) /@ (Sort[#1, Greater] &) /@
Partition[Take[seq, k], 2, 1], n++]; n, {k, 2, Length[seq]}]];
t = Map[leastSeparatorS[1/Range[50], #] &, Range[5]];
TableForm[t]
t[[4]] (* Peter J. C. Moses, Aug 08 2012 *)